2
$\begingroup$

For example, if the convexity radius of a point x in a riemannian manifold M (without boundary) is R, what can we say about the convexity radius of points in B_R(x)?

The convexity radius of x is the sup of R s.t. B_R(x) is convex, and is always positive.

$\endgroup$
1
  • $\begingroup$ Ok, I got some sort of an answer from klingenberg, riemannian geometry, page 85, corollary 1.9.11: the convexity radius (with a different and stronger definition) is lipschitz, with lipschitz constant at most 1! $\endgroup$
    – fritz
    Apr 9, 2014 at 17:01

1 Answer 1

1
$\begingroup$

And the example on page 84 of that book shows that with your definition of convexity we cannot say much. Let $M$ be a very sharp cone, slightly rounded. Like this, only sharper.

enter image description here

Let $x$ be the tip of the cone. Then $B_R(x)$ is convex for all $R$; the convexity radius is infinite. But for all other points $y$, apart from the tiny rounded end, the geodesic ball wraps around the cone before covering the tip. The closer is $y$ to $x$, the sooner this happens (until we enter the tiny rounded end). So the radius of convexity at $y$ can be extremely small.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .